We consider a reflection and transmission dual-functional active RIS, which is employed to assist a MU-MISO wireless communication system as illustrated in Fig. REF . The BS equipped with \(N\)  antennas communicates with \(K\)  single-antenna users with the aid of a dual-functional active RIS. As shown in Fig. REF , some users are located within the same half-space as BS and served by the surface facing them (which is labelled as S-r in Fig. REF ). Meanwhile, other users located at the opposite side are served by S-t.
The set of all users in this system is denoted by \(\mathcal {K}=\lbrace 1,2,\ldots ,K\rbrace \) , while the set of users that receive the signals reflected from the S-r surface is denoted by \(\mathcal {K}_\mathrm {r}\)  and the set of users that receive the signals transmitted from the S-t surface
is denoted by \(\mathcal {K}_\mathrm {t}\) , \(\mathcal {K}_\mathrm {r} \cap \mathcal {K}_\mathrm {t}=\varnothing \)  and \(\mathcal {K}_\mathrm {r} \cup \mathcal {K}_\mathrm {t}=\mathcal {K}\) .
Different from the existing study [1]}, in which the
inter-surface signal reflection is considered and the double RISs are adjusted cooperatively, in this paper, we ignore the multi-hop reflected signals due to the fact that two surfaces of the dual-functional RIS are against each other and the multi-hop reflected signals among surfaces S-r and S-t are weak. Moreover, with the rapid development of meta-surfaces, the interference between reflection and transmission can be controlled to a very low level [2]}, [3]}. Therefore, we assume that the signals emitted from S-t will not be reflected by S-r and received by the users in \(\mathcal {K}_{\mathrm {r}}\)  and vice versa [4]}, [5]}.
By enabling simultaneous reflection and transmission, the dual-functional active RIS extends the service coverage and enhances the strength of signals received by all users, no matter they are located at either side of the RIS.
